Integrand size = 23, antiderivative size = 72 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {\log \left (a+b \tanh ^2(c+d x)\right )}{2 (a+b)^2 d}+\frac {a}{2 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \]
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Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 78} \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {a}{2 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac {\log \left (a+b \tanh ^2(c+d x)\right )}{2 d (a+b)^2}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rule 78
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{(1-x) (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}-\frac {a}{(a+b) (a+b x)^2}+\frac {b}{(a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = \frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {\log \left (a+b \tanh ^2(c+d x)\right )}{2 (a+b)^2 d}+\frac {a}{2 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {2 \log (\cosh (c+d x))+\log \left (a+b \tanh ^2(c+d x)\right )+\frac {a (a+b)}{b \left (a+b \tanh ^2(c+d x)\right )}}{2 (a+b)^2 d} \]
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Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.17
method | result | size |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}-\frac {-\frac {a \left (a +b \right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}-\ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{2 \left (a +b \right )^{2}}}{d}\) | \(84\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 \left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}-\frac {-\frac {a \left (a +b \right )}{b \left (a +b \tanh \left (d x +c \right )^{2}\right )}-\ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{2 \left (a +b \right )^{2}}}{d}\) | \(84\) |
parallelrisch | \(-\frac {2 a b d x -a b +2 \ln \left (1-\tanh \left (d x +c \right )\right ) a b -a^{2}-b^{2} \ln \left (a +b \tanh \left (d x +c \right )^{2}\right ) \tanh \left (d x +c \right )^{2}-\ln \left (a +b \tanh \left (d x +c \right )^{2}\right ) a b +2 x \tanh \left (d x +c \right )^{2} b^{2} d +2 \ln \left (1-\tanh \left (d x +c \right )\right ) \tanh \left (d x +c \right )^{2} b^{2}}{2 \left (a +b \tanh \left (d x +c \right )^{2}\right ) d \left (a +b \right )^{2} b}\) | \(141\) |
risch | \(-\frac {x}{a^{2}+2 a b +b^{2}}-\frac {2 c}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 a \,{\mathrm e}^{2 d x +2 c}}{d \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \left (a^{2}+2 a b +b^{2}\right )}\) | \(159\) |
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Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 629, normalized size of antiderivative = 8.74 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {2 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{4} + 8 \, {\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, {\left (a + b\right )} d x \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + b\right )} d x + 4 \, {\left ({\left (a - b\right )} d x - a\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} + {\left (a - b\right )} d x - a\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \log \left (\frac {2 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{2} + a - b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) + 8 \, {\left ({\left (a + b\right )} d x \cosh \left (d x + c\right )^{3} + {\left ({\left (a - b\right )} d x - a\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right )^{2} + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d + 4 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \cosh \left (d x + c\right )^{3} + {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
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Time = 42.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=- \frac {a \left (\begin {cases} \frac {\tanh ^{2}{\left (c + d x \right )}}{a^{2}} & \text {for}\: b = 0 \\- \frac {1}{b \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )} & \text {otherwise} \end {cases}\right )}{2 d \left (a + b\right )} + \frac {b \left (\begin {cases} \frac {\tanh ^{2}{\left (c + d x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b \tanh ^{2}{\left (c + d x \right )} \right )}}{b} & \text {otherwise} \end {cases}\right )}{2 d \left (a + b\right )^{2}} - \frac {\log {\left (\tanh ^{2}{\left (c + d x \right )} - 1 \right )}}{2 d \left (a + b\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (68) = 136\).
Time = 0.22 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.36 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {2 \, a e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} + 2 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {\log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (68) = 136\).
Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.07 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {\frac {\log \left ({\left | a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2}{{\left (a {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b\right )} {\left (a + b\right )}}}{2 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.92 \[ \int \frac {\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {-a^2+a\,b\,\left (-1+\mathrm {atan}\left (\frac {a\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a-a\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2}\right )\,2{}\mathrm {i}\right )+b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,\mathrm {atan}\left (\frac {a\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a-a\,{\mathrm {tanh}\left (c+d\,x\right )}^2+b\,{\mathrm {tanh}\left (c+d\,x\right )}^2}\right )\,2{}\mathrm {i}}{2\,d\,a^3\,b+2\,d\,a^2\,b^2\,{\mathrm {tanh}\left (c+d\,x\right )}^2+4\,d\,a^2\,b^2+4\,d\,a\,b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^2+2\,d\,a\,b^3+2\,d\,b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^2} \]
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